Hermitian surfaces

Abstract

In this chapter we decribe the l.c.K. surfaces among the Hermitian surfaces. Of course, we are motivated by having at hand examples such as Hopf and Inoue surfaces, which were previously seen to admit natural 1.c.K. structures. On the other hand, in general, on each Hermitian surface the identity

$$ d\Omega = \omega \wedge \Omega $$

holds, where the Lee form ω is given by

$$ \omega = \left( {\delta \Omega } \right) \circ J. $$

Yet, in general, ω is not closed (while, if it satisfies dΩ=ω∧Ω, it is always closed in complex dimension n > 2).